Induced Representation and Frobenius Reciprocity for Compact

Induced Representation and Frobenius Reciprocity for Compact Quantum Groups By ARUPKUMAR PAL Indian Statistical Institute, Delhi Centre 7, SJSS Marg, New Delhi 110 016 e-mail: arup @ isid.ernet.in

Abstract: Unitary representations of compact quantum groups have been described as isometric comodules. The notion of an induced representation for compact quantum groups has been introduced and an analogue of the Frobenius reciprocity theorem is established.

Quantum groups, like their classical counterparts, have a very rich representation theory. In the representation theory of classical groups, induced representation plays a very important role. Among other things, for example, one can obtain families of irreducible unitary representations of many locally compact groups as representations induced by one dimensional representations of appropriate subgroups. Therefore it is natural to try and see how far can this notion be developed and exploited in the case of quantum groups. As a first step, we do it here for compact quantum groups. First we give an alternative description of a unitary representation as an isometric comodule map. This is trivial in the finite dimensional case, but requires a little bit of work if the comodule is infinite dimensional. Using the comodule description, the notion of an induced representation is defined. We then go on to prove that an exact analogue of the Frobenius reciprocity theorem holds for compact quantum groups. As an application of 2 this theorem, an alternative way of decomposing the action of SUq(2) on the Podle`ssphere Sq0 is given. Notations. , etc, with or without subscripts, will denote complex separable Hilbert spaces. H K ( ) and 0( ) denote respectively the space of bounded operators and the space of compact B H B H operators on . , , etc denote C -algebras. All the C -algebras used in this article have H A B C ∗ ∗ been assumed to act nondegenerately on Hilbert spaces. More specifically, given any C∗-algebra , it is assumed that there is a Hilbert space such that ( ) and for u , a(u) = 0 A K A ⊆ B K ∈ K for all a implies u = 0. Tensor product of C -algebras will always mean their spatial ∈ A ∗ tensor product. The identity operator on Hilbert spaces is denoted by I, and on C∗-algebras by id. For two vector spaces X and Y , X alg Y denote their algebraic tensor product. ⊗ 1 Let be a C -algebra acting on . The subalgebras a ( ): ab b and A ∗ K { ∈ B K ∈ A ∀ ∈ A } a ( ): ab, ba b of ( ) are called respectively the left multiplier algebra { ∈ B K ∈ A ∀ ∈ A } B K and the multiplier algebra of . We denote them by LM( ) and M( ) respectively. A good A A A reference for multiplier algebras and other topics in C∗-algebra theory is [4]. See [9] for another equivalent description of multiplier algebras that is often very useful.

1 Preliminaries

1.1 Let be a unital C -algebra. A vector space X having a right -module structure is A ∗ A called a Hilbert -module if it is equipped with an -valued inner product that satisfies A A i. x, y = y, x , h i∗ h i ii. x, x 0, h i ≥ iii. x, x = 0 x = 0, h i ⇒ iv. x, yb = x, y b for x, y X, b , h i h i ∈ ∈ A and if x := x, x 1/2 makes X a Banach Space. k k kh ik Details on Hilbert C∗-modules can be found in [1], [2] and [3]. We shall need a few specific examples that are listed below. Examples(a) Any Hilbert space with its usual inner product is a Hilbert C/ -module. H (b) Any unital C -algebra with a, b = a b is a Hilbert -module. ∗ A h i ∗ A (c) , the ‘external tensor product’ of and , is a Hilbert -module. H ⊗ A H A A (d) ( , ), with S, T = S T is a Hilbert ( )-module. B H K h i ∗ B H 1.2 We have seen above that ( ) and ( , ) both are Hilbert ( )-modules. It is H ⊗ B K B K H ⊗ K B K easy to see that the map ϑ : ui ai ui ai(.) from alg ( ) to ( , ) extends P ⊗ 7→ P ⊗ H ⊗ B K B K H ⊗ K to an isometric module map from ( ) to ( , ), i.e. ϑ obeys H ⊗ B K B K H ⊗ K ϑ(x), ϑ(y) = x, y , x, y ( ), h i h i ∀ ∈ H ⊗ B K ϑ(xb) = ϑ(x)b, x ( ), b ( ). ∀ ∈ H ⊗ B K ∈ B K Thus ϑ embeds ( ) in ( , ). Observe two things here: first, if = C/ , ϑ is just H ⊗ B K B K H ⊗ K H the identity map. And, ϑ is onto if and only if is finite dimensional. The following lemma, H the proof of which is fairly straightforward, gives a very useful property of ϑ.

Lemma Let ϑi be the map ϑ constructed above with i replacing , i = 1, 2. Let S ( 1, 2) H H ∈ B H H and x 1 ( ). Then ϑ2((S id)x) = (S I)ϑ1(x). ∈ H ⊗ B K ⊗ ⊗

2 1.3 For an operator T ( ), and a vector u , let Tu denote the operator v T (u v) ∈ B H⊗K ∈ H 7→ ⊗ from to . It is not too diﬃcult to show that Tu ϑ( ( )) if T LM( 0( ) ( )). K H⊗K ∈ H⊗B K ∈ B H ⊗B K 1 Deﬁne a map Ψ(T ) from to ( ) by: Ψ(T )(u) = ϑ (Tu). Then Ψ is the unique linear H H ⊗ B K − injective contraction from LM( 0( ) ( )) to ( , ( )) for which ϑ(Ψ(T )(u))(v) = B H ⊗ B K B H H ⊗ B K T (u v) u , v , T LM( 0( ) ( )). Here are a few interesting properties of this ⊗ ∀ ∈ H ∈ K ∈ B H ⊗ B K map Ψ.

Proposition Let Ψ: LM( 0( ) ( )) ( , ( )) be the map described above. B H ⊗ B K → B H H ⊗ B K Then we have the following: i. Ψ maps isometries in LM( 0( ) ( )) onto the isometries in ( , ( )). B H ⊗ B K B H H ⊗ B K ii. For any T LM( 0( ) ( )) and S 0( ), ∈ B H ⊗ B K ∈ B H Ψ(T (S I)) = Ψ(T ) S, Ψ((S I)T ) = (S id) Ψ(T ). ⊗ ◦ ⊗ ⊗ ◦ iii. If is any C -subalgebra of ( ) containing its identity, then T LM( 0( ) ) if and A ∗ B K ∈ B H ⊗ A only if range Ψ(T ) . ⊆ H ⊗ A Proof : i. Suppose T LM( 0( ) ( )) is an isometry. By 1.2, Ψ(T )u, Ψ(T )v = ∈ B H ⊗ B K h i 1 1 ϑ (Tu), ϑ (Tv) = Tu,Tv = u, v I for u, v . Thus Ψ(T ) is an isometry. h − − i h i h i ∈ H Conversely, take an isometry π : ( ) and deﬁne an operator T on the product H → H ⊗ B K vectors in by T (u v) = ϑ(π(u))(v), ϑ being the map constructed in 1.2. It is clear that H ⊗ K ⊗ T is an isometry. It is enough, therefore, to show that T ( u v S) 0( ) ( ) whenever | ih | ⊗ ∈ B H ⊗ B K S ( ) and u, v are unit vectors in such that u, v = 0 or 1. ∈ B K H h i Choose an orthonormal basis ei for such that e1 = u, er = v where { } H 0 if u, v = 0, r =  h i  1 if u, v = 1.  h i Let πij = ( ei id)π(ej). Then T ( u v S) = ei er πi1S where the right hand side h | ⊗ | ih | ⊗ P | ih | ⊗ converges strongly. Since π(e1) ( ), it follows that πi1∗πi1 converges in norm. ∈ H ⊗ B K Pi Consequently the right hand side above converges in norm, which means T ( u v S) | ih | ⊗ ∈ 0( ) ( ). B H ⊗ B K ii. Straightforward. iii. Take T = u v a, u, v , a . For any w , Ψ(T )(w) = v, w u a . | ih | ⊗ ∈ H ∈ A ∈ H h i ⊗ ∈ H ⊗ A Since Ψ is a contraction, and the norm closure of all linear combinations of such T ’s is 0( ) , B H ⊗A we have range Ψ(T ) for all T 0( ) . ⊆ H ⊗ A ∈ B H ⊗ A Assume next that T LM( 0( ) ). Then T ( u u I) 0( ) for all u . ∈ B H ⊗ A | ih | ⊗ ∈ B H ⊗ A ∈ H Hence Ψ(T ( u u I))(u) , which means, by part (ii), that Ψ(T )(u) for all | ih | ⊗ ∈ H ⊗ A ∈ H ⊗ A u . Thus range Ψ(T ) . ∈ H ⊆ H ⊗ A 3 To prove the converse, it is enough to show that T ( u v a) 0( ) whenever a | ih | ⊗ ∈ B H ⊗ A ∈ A and u, v are such that u, v = 0 or 1. Rest of the proof goes along the same lines as the ∈ H h i proof of the last part of (i).

1.4 Let 1, 2 be two Hilbert spaces, i being a C -subalgebra of ( i) containing its identity. K K A ∗ B K Suppose φ is a unital *-homomorphism from 1 to 2. Then id φ : S a S φ(a) extends A A ⊗ ⊗ 7→ ⊗ to a *-homomorphism from 0( ) 1 to 0( ) 2. Moreover ((id φ)(a))b : a B H ⊗ A B H ⊗ A { ⊗ ∈ 0( ) 1, b 0( ) 2 is total in 0( ) 2. Therefore id φ extends to an algebra B H ⊗ A ∈ B H ⊗ A } B H ⊗ A ⊗ homomorphism by the following prescription: for all a LM( 0( ) 1), b 0( ) 1, ∈ B H ⊗ A ∈ B H ⊗ A c 0( ) 2, ∈ B H ⊗ A ((id φ)a)(((id φ)b)c) := ((id φ)(ab))c. ⊗ ⊗ ⊗ Proposition Let φ be as above, and Ψi be the map Ψ constructed earlier with i replacing . K K Then for T LM( 0( ) 1), ∈ B H ⊗ A

(I φ)Ψ1(T ) = Ψ2((id φ)T ). ⊗ ⊗

Proof : It is enough to prove that ( u id)((I φ)Ψ1(T )(v)) = ( u id)Ψ2((id φ)T )(v), h | ⊗ ⊗ h | ⊗ ⊗ u, v . Rest now is a careful application of 1.2. ∀ ∈ H

1.5 Consider the homomorphic embeddings φ12 : 0( ) 1 0( ) 1 2 and φ13 : B H ⊗ A → B H ⊗ A ⊗ A 0( ) 2 0( ) 1 2 given on the product elements by B H ⊗ A → B H ⊗ A ⊗ A

φ12(a b) = a b I, φ13(a c) = a I c ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ respectively. Each of their ranges contains an approximate identity for 0( ) 1 2, so that B H ⊗A ⊗A their extensions respectively to LM( 0( ) 1) and LM( 0( ) 2) are also homomorphic B H ⊗ A B H ⊗ A embeddings.

Proposition Let Ψ1, Ψ2 be as in the previous proposition, and let Ψ0 be the map Ψ with

1 2 replacing . Let S LM( 0( ) 1), T LM( 0( ) 2). Then A ⊗ A A ∈ B H ⊗ A ∈ B H ⊗ A

Ψ0(φ12(S)φ13(T )) = (Ψ1(S) id)Ψ2(T ). ⊗

2 Proof : Observe that for u1, . . . , un , (( Ψ1(S)(ui), Ψ1(S)(uj) )) S (( ui, uj I)). There- ∈ H h i ≤ k k h i fore Ψ1(S) id is a well-deﬁned bounded operator from 2 to 1 2. Take an ⊗ H ⊗ A H ⊗ A ⊗ A orthonormal basis ei for . Deﬁne Sij’s and Tij’s as follows: { } H

Sij : v ( ei I)S(ej v),Tij : v ( ei I)T (ej v). 7→ h | ⊗ ⊗ 7→ h | ⊗ ⊗ 4 n Let Pn := i=1 ei ei . Then (Ψ1(S) id)(Pn id)Ψ2(T )(ei) = (Ψ1(S) id)( j n ej Tij) = P | ih | ⊗ ⊗ ⊗ P ≤ ⊗ j n( k ek Skj) Tij. Hence for v 1, w 2, P ≤ P ⊗ ⊗ ∈ K ∈ K

ϑ((Ψ1(S) id)(Pn id)Ψ2(T )(ei))(v w) ⊗ ⊗ ⊗ = e S (v) T (w) X X k kj ji j n k ⊗ ⊗ ≤ = ( e e S T )(e v w) X X k r kj ji i j n k,r | ih | ⊗ ⊗ ⊗ ⊗ ≤ = φ12(S)(Pn I I)φ13(T )(ei v w). ⊗ ⊗ ⊗ ⊗

This converges to φ12(S)φ13(T )(ei v w) as n . On the other hand, ⊗ ⊗ → ∞

lim (Ψ1(S) id)(Pn id)Ψ2(T )(ei) = (Ψ1(S) id)Ψ2(T )(ei), n →∞ ⊗ ⊗ ⊗ which implies limn ϑ((Ψ1(S) id)(Pn id)Ψ2(T )(ei)) = ϑ((Ψ1(S) id)Ψ2(T )(ei)). Therefore →∞ ⊗ ⊗ ⊗ ϑ((Ψ1(S) id)Ψ2(T )(ei))(v w) = φ12(S)φ13(T )(ei v w) = ⊗ ⊗ ⊗ ⊗ ϑ(Ψ0(φ12(S)φ13(T ))(ei))(v w). Thus (Ψ1(S) id)Ψ2(T ) = Ψ0(φ12(S)φ13(T )). ⊗ ⊗

2 Representations of Compact Quantum Groups

2.1 We start by recalling a few facts from [8] on compact quantum groups. Deﬁnition Let be a separable unital C -algebra, and µ : be a unital *- A ∗ A → A ⊗ A homomorphism. We call G = ( , µ) a compact quantum group if the following two conditions A are satisﬁed: i. (id µ)µ = (µ id)µ, and ⊗ ⊗ ii. (a I)µ(b): a, b and (I a)µ(b): a, b both are total in . { ⊗ ∈ A} { ⊗ ∈ A} A ⊗ A µ is called the comultiplication map associated with G. We shall very often denote the underlying C -algebra by C(G) and the map µ by µG. ∗ A A representation of a compact quantum group G acting on a Hilbert space is an element H π of the multiplier algebra M( 0( ) C(G)) that obeys π12π13 = (id µ)π, where π12 and B H ⊗ ⊗ π13 are the images of π in the space M( 0( ) C(G) C(G)) under the homomorphisms φ12 B H ⊗ ⊗ and φ13 which are given on the product elements by:

φ12(a b) = a b I, φ13(a b) = a I b. ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

A representation π is called a unitary representation if ππ∗ = I = π∗π. One also has the notions of irreducibility, direct sum and tensor product of representations. As in the case of

5 classical groups, any unitary representation decomposes into a direct sum of ﬁnite dimensional irreducible unitary representations. Let A(G) be the unital -subalgebra of C(G) generated ∗ by the matrix entries of ﬁnite dimensional unitary representations of G. Then one has the following result (see [8]). Theorem ([8]) Suppose G is a compact quantum group. Let A(G) be as above. Then we have the following:

(a) A(G) is a dense unital *-subalgebra of C(G) and µ(A(G)) A(G) alg A(G). ⊆ ⊗ (b) There is a complex homomorphism : A(G) C/ such that → ( id)µ = id = (id )µ. ⊗ ⊗ (c) There exists a linear antimultiplicative map κ : A(G) A(G) obeying →

m(id κ)µ(a) = (a)I = m(κ id)µ(a), and κ(κ(a∗)∗) = a ⊗ ⊗ for all a A(G), where m is the operator that sends a b to ab. ∈ ⊗ The maps and κ in the above theorem are called the counit and coinverse respectively of the quantum group G.

2.2 Let G = (C(G), µG) and H = (C(H), µH ) be two compact quantum groups. A C∗- homomorphism φ from C(G) to C(H) is called a quantum group homomorphism from G to H if it obeys (φ φ)µG = µH φ. ⊗ One can show that if G, H are compact quantum groups, then H is a subgroup of G if and only if there is a homomorphism from G to H that maps C(G) onto C(H).

2.3 Let G = ( , µ) be a compact quantum group. From now onward we shall assume that A acts nondegenerately on a Hilbert space , i.e. is a C -subalgebra of ( ) containing A K A ∗ B K its identity. We call a map π from to an isometry if π(u), π(v) = u, v I for all H H ⊗ A h i h i u, v . If π : is an isometry, then π id : u a π(u) a extends to a bounded ∈ H H → H ⊗ A ⊗ ⊗ 7→ ⊗ map from to . π is called an isometric comodule map if it is an isometry, and H ⊗ A H ⊗ A ⊗ A satisﬁes (π id)π = (I µ)π. The pair ( , π) is called an isometric comodule. We shall often ⊗ ⊗ H just say π is a comodule, omitting the . H The following theorem says that for a compact quantum group isometric comodules are nothing but the unitary representations. Theorem Let π be an isometric comodule map acting on . Then Ψ 1(π) is a unitary repre- H − sentation acting on . Conversely, if πˆ is a unitary representation of G on , then ( , Ψ(ˆπ)) H H H is an isometric comodule.

6 We need the following lemma for proving the theorem. Lemma Let ( , π) be an isometric comodule. Then decomposes into a direct sum of ﬁnite H H dimensional subspaces = α such that each α is π-invariant and π α is an irreducible H ⊕H H |H isometric comodule.

Proof : By 1.3, there is an isometryπ ˆ in LM( 0( ) ) such that Ψ(ˆπ) = π. Using 1.4 and B H ⊗ A 1.5, we getπ ˆ12πˆ13 = (id µ)ˆπ whereπ ˆ12 = φ12(ˆπ),π ˆ13 = φ13(ˆπ), φ12 and φ13 being as in 1.5 ⊗ with 1 = 2 = . A A A Let = a : h(a a) = 0 . From the properties of the haar state, is an ideal in . I { ∈ A ∗ } I A For any unit vector u in , let Q(u) = (id h)(ˆπ( u u I)ˆπ ). Then Q(u) = Q(u) 0( ). H ⊗ | ih | ⊗ ∗ ∗ ∈ B H 1/2 If Q(u) = 0, then πˆ( u u I)ˆπ 0( ) . Thereforeπ ˆ( u u I)ˆπ 0( ) . | | ih | ⊗ ∗| ∈ B H ⊗ I | ih | ⊗ ∗ ∈ B H ⊗ I It follows then that u u I 0( ) . This forces u to be zero. Thus for a nonzero u, | ih | ⊗ ∈ B H ⊗ I Q(u) = 0. Choose and fix any nonzero u. Then 6 πˆ(Q(u) I)ˆπ∗ ⊗ = (id id h)(ˆπ12πˆ13( u u I I)ˆπ∗ πˆ∗ ) ⊗ ⊗ | ih | ⊗ ⊗ 13 12 = (id id h)(ˆπ12πˆ13( u u I I) (ˆπ12πˆ13( u u I I))∗) ⊗ ⊗ | ih | ⊗ ⊗ | ih | ⊗ ⊗ = (id id h)((id µ)(ˆπ)(id µ)( u u I) ((id µ)(ˆπ)(id µ)( u u I))∗) ⊗ ⊗ ⊗ ⊗ | ih | ⊗ ⊗ ⊗ | ih | ⊗ = (id id h)((id µ)(ˆπ( u u I)) ((id µ)(ˆπ( u u I)))∗) ⊗ ⊗ ⊗ | ih | ⊗ ⊗ | ih | ⊗ = (id id h)((id µ)(ˆπ( u u I)) (id µ)(( u u I)ˆπ∗)) ⊗ ⊗ ⊗ | ih | ⊗ ⊗ | ih | ⊗ = (id id h)(id µ)(ˆπ( u u I)ˆπ∗) ⊗ ⊗ ⊗ | ih | ⊗ = (id (id h)µ)(ˆπ( u u I)ˆπ∗) ⊗ ⊗ | ih | ⊗ = Q(u) I. ⊗ Thusπ ˆ(Q(u) I) = (Q(u) I)ˆπ. If P is any finite dimensional spectral projection of Q(u), then ⊗ ⊗ πˆ(P I) = (P I)ˆπ, which means, by an application of part (ii) of 1.3, that πP = (P id)π. ⊗ ⊗ ⊗ Standard arguments now tell us that π can be decomposed into a direct sum of finite dimensional isometric comodules. Finite dimensional comodules, in turn, can easily be shown to decompose into a direct sum of irreducible isometric comodules. The proof is thus complete. Proof of the theorem: Letπ ˆ be a unitary representation. By 1.3, Ψ(ˆπ) is an isometry from to C(G). Using 1.4 and 1.5, we conclude that Ψ(ˆπ) is an isometric comodule. H H ⊗ For the converse, take an isometric comodule π. If π is finite dimensional, it is easy to 1 see that Ψ− (π) is a unitary representation. So assume that π is infinite dimensional. By the lemma above, there is a family Pα of finite dimensional projections in ( ) satisfying { } B H

PαPβ = δαβPα, Pα = I, πPα = (Pα id)π α (2.1) X ⊗ ∀ 7 such that π Pα = πPα is an irreducible isometric comodule. π Pα is ﬁnite dimensional, | H | H 1 therefore Ψ− (π Pα ) is a unitary element of LM( 0(Pα ) ) = (Pα ) . Let us denote | H B H ⊗A B H ⊗A Ψ 1(π) byπ ˆ. Then the above implies that in the bigger space ( ), − B H ⊗ K

(ˆπ(Pα I))∗(ˆπ(Pα I)) = Pα I = (ˆπ(Pα I))(ˆπ(Pα I))∗. ⊗ ⊗ ⊗ ⊗ ⊗

The second equality implies thatπ ˆ(Pα I)ˆπ = Pα I for all α, so thatπ ˆπˆ = I. We already ⊗ ∗ ⊗ ∗ know, by 1.3 thatπ ˆ πˆ = I and by 1.4 and 1.5 thatπ ˆ12πˆ13 = (id µ)ˆπ. Thus it remains only ∗ ⊗ to show thatπ ˆ M( 0( ) ). It is enough to show that for any S 0( ) and a , ∈ B H ⊗ A ∈ B H ∈ A (S a)ˆπ 0( ) . Now from (2.1) and 1.3,π ˆ(Pα I) = (Pα I)ˆπ for all α. Therefore ⊗ ∈ B H ⊗ A ⊗ ⊗ (S a)(Pα I)ˆπ = (S a)ˆπ(Pα I) 0( ) . Since (S a)ˆπ is the norm limit of ﬁnite ⊗ ⊗ ⊗ ⊗ ∈ B H ⊗ A ⊗ sums of such terms, (S a)ˆπ 0( ) . Thusπ ˆ is a unitary representation acting on . ⊗ ∈ B H ⊗ A H

2.4 Next we introduce the right regular comodule. Denote by L2(G) the GNS space associated with the haar state h on G. Then is a dense subspace of L2(G). One can also see that A A ⊗ A can be regarded as a subspace of L2(G) . Consider the map µ : . ⊗ A A → A ⊗ A

µ(a), µ(b) = (h id)(µ(a)∗µ(b)) = (h id)µ(a∗b) = h(a∗b)I = a, b I h i ⊗ ⊗ h i for all a, b . Therefore µ extends to an isometry from L2(G) into L2(G) . Denote it ∈ A ⊗ A by . The maps (I µ) and ( id) both are isometries from L2(G) to L2(G) < ⊗ < < ⊗ < ⊗ A ⊗ A and they coincide on . Hence (I µ) = ( id) . Thus is an isometric comodule map. A ⊗ < < ⊗ < < We call it the right regular comodule of G. By theorem 2.3, Ψ 1( ) is a unitary representation − < acting on L2(G). This is the right regular representation introduced by Woronowicz ([8]). Finally let us state here a small lemma which is a direct consequence of the Peter-Weyl theorem for compact quantum groups.

2.5 Lemma u L2(G): (u) L2(G) alg C(G) = A(G). { ∈ < ∈ ⊗ }

3 Induced Representations

In this section we shall introduce the concept of an induced representation and show that Frobenius reciprocity theorem holds for compact quantum groups. Throughout this section

G = (C(G), µG) will denote a compact quantum group and H = (C(H), µH ), a subgroup of G. We start with a lemma concerning the boundedness of the left convolution operator.

3.1 Lemma Let G = ( , µ) be a compact quantum group. Then the map Lρ : given A A → A by Lρ(a) = (ρ id)µ(a) extends to a bounded operator from L2(G) into itself. ⊗ 8 Proof : The proof follows from the following inequality: for any two states ρ1 and ρ2 on , we A have

ρ1((ρ2 a)∗(ρ2 a)) ρ2 ρ1(a∗a) a , ∗ ∗ ≤ ∗ ∀ ∈ A where ρi a := (ρi id)µ(a). ∗ ⊗ 3.2 Letπ ˆ be a unitary representation of H acting on the space 0. π := Ψ(ˆπ) is then an H isometric comodule map from 0 to 0 C(H). Consider the following map from 0 L2(G) H H ⊗ H ⊗ to 0 L2(G) C(G): H ⊗ ⊗ I G : u v u G(v) ⊗ < ⊗ 7→ ⊗ < where G is the right regular comodule of G. It is easy to see that this is an isometric comodule < map acting on 0 L2(G). H ⊗ Let p be the homomorphism from G to H (cf. 2.2). Let = u 0 L2(G): H { ∈ H ⊗ G (I Lρ p)u = (πρ I)u for all continuous linear functionals ρ on C(H) . Then I ⊗ · ⊗ } ⊗ < keeps invariant; the restriction of I G to is therefore an isometric comodule, so that H ⊗ < H 1 G Ψ− ((I ) ) is a unitary representation of G acting on . We call this the representation ⊗ < |H H G induced byπ ˆ, and denote it by indH πˆ or simply by indπ ˆ when there is no ambiguity about G and H.

Letπ ˆ1 andπ ˆ2 be two unitary representations of H. Then clearly we have i. indπ ˆ1 and indπ ˆ2 are equivalent wheneverπ ˆ1 andπ ˆ2 are equivalent, and ii. ind (ˆπ1 πˆ2) and indπ ˆ1 indπ ˆ2 are equivalent. ⊕ ⊕ Before going to the Frobenius reciprocity theorem, let us brieﬂy describe what we mean by restriction of a representation to a subgroup. Letπ ˆG be a unitary representation of G acting G G G H on a Hilbert space 0. We call (id p)ˆπ the restriction ofπ ˆ to H and denote it byπ ˆ . H ⊗ | To see that it is indeed a unitary representation, observe that Ψ((id p)ˆπG) = (I p)Ψ(ˆπG) ⊗ ⊗ which is clearly an isometric comodule. Therefore by 2.3,π ˆG H is a unitary representation of | G G G H G H H acting on 0. Denote Ψ(ˆπ ) by π and Ψ(ˆπ ) by π . H | | 3.3 Theorem Let πˆG and πˆH be irreducible unitary representations of G and H respectively. Then the multiplicity of πˆG in indG πˆH is the same as that of πˆH in πˆG H . H |

Proof : Let (ˆπG H , πˆH ) (respectively (ˆπG, indπ ˆH )) denote the space of intertwiners between I | I G H H G H G H πˆ andπ ˆ (respectivelyπ ˆ and indπ ˆ ). Assume thatπ ˆ andπ ˆ act on 0 and 0 | K H G respectively. 0 C(G) can be regarded as a subspace of 0 L2(G) and hence π , as a map K ⊗ K ⊗ G G from 0 into 0 L2(G). Since π = Ψ(ˆπ ) is unitary, we have for u, v 0, K K ⊗ ∈ K G G G G π (u), π (v) 0 L2(G) = h( π (u), π (v) 0 C(G)) = h( u, v I) = u, v . h iK ⊗ h iK ⊗ h i h i 9 G G H H Thus π : 0 0 L2(G) is an isometry. Let S : 0 0 be an element of (ˆπ , πˆ ). K → K ⊗ K → H I | G (S I)π is then a bounded map from 0 into 0 L2(G). Denote it by f(S). It is not too ⊗ K H ⊗ G H diﬁcult to see that f(S) actually maps 0 into , and intertwinesπ ˆ and indπ ˆ . f : S f(S) K H 7→ is thus a linear map from (ˆπG H , πˆH ) to (ˆπG, indπ ˆH ). I | I We shall now show that f is invertible by exhibiting the inverse of f. Take a T : 0 K → H G H u that intertwinesπ ˆ and indπ ˆ . For any u 0, T := ( u I)T is a map from 0 to L2(G) ∈ H h | ⊗ K intertwiningπ ˆG and the right regular representation G of G, i.e. GT u = (T u id)πG. Now, < < ⊗ G G G u π is ﬁnite dimensional, so that π ( 0) 0 alg A(G). Hence T ( 0) L2(G) alg A(G). K ⊆ K ⊗ < K ⊆ ⊗ u By 2.5, T ( 0) A(G). Since this is true for all u 0, T ( 0) 0 alg A(G). Therefore K ⊆ ∈ H K ⊆ H ⊗ (I G)T is a bounded operator from 0 to 0. Denote it by g(T ). ⊗ K H For a comodule π and a linear functional ρ, denote (id ρ)π by πρ. Let ρ be a linear ⊗ H H H functional on C(H). Then πρ g(T ) = πρ (I G)T = (I G)(πρ id)T = (I G)(I Lρ p)T = ⊗ ⊗ ⊗ ⊗ ⊗ · G H G H (I ρ p)T . On the other hand, since T intertwinesπ ˆ and indπ ˆ , we have g(T )(π )ρ = ⊗ ◦ | G H G G G g(T )(I ρ)π = g(T )(I ρ)(I p)π = (I G)T πρ p = (I G)(I ρ p)T = (I ρ p)T . ⊗ | ⊗ ⊗ ⊗ · ⊗ ⊗ < · ⊗ ◦ H G H Thus π g(T ) = g(T )(π )ρ for all continuous linear functionals ρ on C(H), which implies ρ | g(T ) (ˆπG H , πˆH ). The map T g(T ) is the inverse of f. Therefore (ˆπG H , πˆH ) = ∈ I | 7→ I | ∼ (ˆπG, indπ ˆH ), which proves the theorem. I Corollary 1. For any unitary representation πˆG of G and πˆH of H, the spaces (ˆπG H , πˆH ) I | and (ˆπG, ind πˆH ) are isomorphic. I Corollary 2. Let H be a subgroup of G and K be a subgroup of H. Suppose πˆ is a unitary G G H representation of K. Then indK πˆ and indH (indK πˆ) are equivalent.

2 3.4 Action of SUq(2) on the sphere Sq0 has been decomposed by Podle`s(see[5]). Here we give an alternative way of doing it using the Frobenius reciprocity theorem. Let us start with a few observations. Let u be the function z z, z S1, where S1 is 7→ ∈ 1 the unit circle in the complex plane. Then u is unitary, and generates the C∗-algebra C(S ) of continuous functions on S1. Let α and β be the two elements that generate the algebra

C(SUq(2)) and obey the following relations:

2 α∗α + β∗β = I = αα∗ + q ββ∗,

αβ qβα = 0 = αβ∗ qβ∗α, β∗β = ββ∗ − − 1 The map p : α u, β 0 extends to a C -homomorphism from C(SUq(2)) onto C(S ). It 7→ 7→ ∗ 1 is in fact a quantum group homomorphism. By 2.2, S is a subgroup of SUq(2).

10 For any n 0, 1/2, 1, 3/2,... , if we restrict the right regular comodule of SUq(2) to ∈ { } < the subspace n of L2(SUq(2)) spanned by H i 2n i α∗ β − : i = 0, 1,..., 2n , (3.1) { } then we get an irreducible isometric comodule. Denote it by u(n). It is a well-known fact ([7]) that these constitute all the irreducible comodules of SUq(2). If we take the basis of n to be H (3.1) with proper normalization, the matrix entries of u(n) turn out to be

(2n j) i (n) (n) (n) 1/2 − ∧  i   2n i  r r(2i r+1)+(j i)(2n j) uij = (di /dj ) − ( 1) q − − − X r r + j i − r=(i j) 0   2   2 − ∨ q− − q−

i r 2n j r r+j i r α − α β β , × ∗ − − − ∗ where k 2 (n)  k  r r(2k r+1) 1 q d = ( 1) q − − ; k X r − 1 q4n+2r 2k+2 r=0   2 − q− −

r (r) 2 (r 1) 2 ... (1) 2   := q− − q− q− ; s (s)q 2 (s 1)q 2 ... (1)q 2 (r s)q 2 (r s 1)q 2 ... (1)q 2   2 − − − − − − q− − − − − 2 4 2k+2 (k) 2 := 1 + q + q + ... + q . q− − − − Since u(n) S1 = (I p)u(n), matrix entries of u(n) S1 are given by | ⊗ | 2(n i) (n) S1  u − if i = j, (u )ij = (3.2) |  0 if i = j.  6 Therefore if n is an integer then the trivial representation occurs in u(n) S1 with multiplicity 1, | and does not occur otherwise. 2 2 Consider now the action of SUq(2) on S . Recall ([5]) that C(S ) = a C(SUq(2)) : q0 q0 { ∈ (p id)µ(a) = I a and the action is the restriction of µ to C(S2 ). From the above ⊗ ⊗ } q0 2 2 description, C(Sq0) can easily be shown to be equal to a C(Sq0): Lρ p(a) = ρ(I)a for all { ∈ · continuous linear functionals ρ on C(S1) . Therefore when we take the closure of C(S2 ) } q0 with respect to the invariant inner product that it carries and extend the action there as an isometry, what we get is the restriction of the right regular comodule of SUq(2) to the subspace < 1 = u L2(SUq(2)) : Lρ p(u) = ρ(I)u for all continuous linear functionals ρ on C(S ) , H { ∈ · } 1 which is nothing but the representationπ ˆ of SUq(2) induced by the trivial representation of S on C/ . Hence the multiplicity of u(n) inπ ˆ is same as that of the trivial representation of S1 in

11 u(n) S1 which is, from (3.2), 1 if n is an integer and 0 if n is not. Thus the action splits into a | direct sum of all the integer-spin representations.

Acknowledgement. I express my gratitude to Prof K. R. Parthasarathy for suggesting the problem and for many useful discussions that I have had with him.

References 1. Jensen, K.K. & Thomsen, K. : Elements of KK-Theory, Birkh¨auser, 1991.

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3. Paschke, W. : Innner Product Modules Over B∗-algebras, Trans. Amer. Math. Soc., 182(1973), 443–468.

4. Pedersen, G.K. : C∗-algebras and Their Automorphism Groups, Academic Press, 1979.

5. Podle`s,P. : Quantum Spheres, Lett. Math. Phys., 14(1987), 193–202.

6. Woronowicz, S.L. : Twisted SU(2) Group. An Example of a Noncommutative Diﬀerential Calculus, Publ. RIMS, Kyoto Univ, 23(1987), 117–181.

7. Woronowicz, S.L. : Compact Matrix Pseudogroups, Comm. Math. Phys., 111(1987), 613–665.

8. Woronowicz, S.L. : Compact Quantum Groups, Preprint, 1992.

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12 Induced Representation and Frobenius Reciprocity for Compact Quantum Groups By ARUPKUMAR PAL Indian Statistical Institute, Delhi Centre 7, SJSS Marg, New Delhi 110 016 e-mail: arup @ isid.ernet.in

August 26, 1994

Key Words. Induced representation, compact quantum group, Hilbert C∗-module.

Running Title. Induced Representation for Quantum Groups.